LMFDB - Embedded newform 930.2.a.n.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [930,2,Mod(1,930)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("930.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(930, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 930 = 2 \cdot 3 \cdot 5 \cdot 31 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	930.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,1,1,1,-1,1,2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$7.42608738798$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	930.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$

$=$

\(q+1.00000 q^{2} +1.00000 q^{3} +1.00000 q^{4} -1.00000 q^{5} +1.00000 q^{6} +2.00000 q^{7} +1.00000 q^{8} +1.00000 q^{9} -1.00000 q^{10} +1.00000 q^{12} -4.00000 q^{13} +2.00000 q^{14} -1.00000 q^{15} +1.00000 q^{16} +6.00000 q^{17} +1.00000 q^{18} +8.00000 q^{19} -1.00000 q^{20} +2.00000 q^{21} +1.00000 q^{24} +1.00000 q^{25} -4.00000 q^{26} +1.00000 q^{27} +2.00000 q^{28} -1.00000 q^{30} +1.00000 q^{31} +1.00000 q^{32} +6.00000 q^{34} -2.00000 q^{35} +1.00000 q^{36} -4.00000 q^{37} +8.00000 q^{38} -4.00000 q^{39} -1.00000 q^{40} -6.00000 q^{41} +2.00000 q^{42} +8.00000 q^{43} -1.00000 q^{45} -12.0000 q^{47} +1.00000 q^{48} -3.00000 q^{49} +1.00000 q^{50} +6.00000 q^{51} -4.00000 q^{52} -6.00000 q^{53} +1.00000 q^{54} +2.00000 q^{56} +8.00000 q^{57} -6.00000 q^{59} -1.00000 q^{60} +2.00000 q^{61} +1.00000 q^{62} +2.00000 q^{63} +1.00000 q^{64} +4.00000 q^{65} +2.00000 q^{67} +6.00000 q^{68} -2.00000 q^{70} -6.00000 q^{71} +1.00000 q^{72} +8.00000 q^{73} -4.00000 q^{74} +1.00000 q^{75} +8.00000 q^{76} -4.00000 q^{78} +8.00000 q^{79} -1.00000 q^{80} +1.00000 q^{81} -6.00000 q^{82} +12.0000 q^{83} +2.00000 q^{84} -6.00000 q^{85} +8.00000 q^{86} -1.00000 q^{90} -8.00000 q^{91} +1.00000 q^{93} -12.0000 q^{94} -8.00000 q^{95} +1.00000 q^{96} -10.0000 q^{97} -3.00000 q^{98} +O(q^{100})\)

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	1.00000	0.500000
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	1.00000	0.408248
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	1.00000	0.353553
$9$	1.00000	0.333333
$10$	−1.00000	−0.316228
$11$	0	0		−	1.00000i	$-0.5\pi$
$11$	0	0			1.00000i	$0.5\pi$
$12$	1.00000	0.288675
$13$	−4.00000	−1.10940	−0.554700	−	0.832050i	$-0.687167\pi$
$13$	−4.00000	−1.10940	−0.554700	+	0.832050i	$0.687167\pi$
$14$	2.00000	0.534522
$15$	−1.00000	−0.258199
$16$	1.00000	0.250000
$17$	6.00000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	6.00000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	1.00000	0.235702
$19$	8.00000	1.83533	0.917663	−	0.397360i	$-0.130073\pi$
$19$	8.00000	1.83533	0.917663	+	0.397360i	$0.130073\pi$
$20$	−1.00000	−0.223607
$21$	2.00000	0.436436
$22$	0	0
$23$	0	0		−	1.00000i	$-0.5\pi$
$23$	0	0			1.00000i	$0.5\pi$
$24$	1.00000	0.204124
$25$	1.00000	0.200000
$26$	−4.00000	−0.784465
$27$	1.00000	0.192450
$28$	2.00000	0.377964
$29$	0	0		−	1.00000i	$-0.5\pi$
$29$	0	0			1.00000i	$0.5\pi$
$30$	−1.00000	−0.182574
$31$	1.00000	0.179605
$31$	1.00000	0.179605
$32$	1.00000	0.176777
$33$	0	0
$34$	6.00000	1.02899
$35$	−2.00000	−0.338062
$36$	1.00000	0.166667
$37$	−4.00000	−0.657596	−0.328798	−	0.944400i	$-0.606644\pi$
$37$	−4.00000	−0.657596	−0.328798	+	0.944400i	$0.606644\pi$
$38$	8.00000	1.29777
$39$	−4.00000	−0.640513
$40$	−1.00000	−0.158114
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	2.00000	0.308607
$43$	8.00000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	8.00000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	0	0
$45$	−1.00000	−0.149071
$46$	0	0
$47$	−12.0000	−1.75038	−0.875190	−	0.483779i	$-0.839264\pi$
$47$	−12.0000	−1.75038	−0.875190	+	0.483779i	$0.839264\pi$
$48$	1.00000	0.144338
$49$	−3.00000	−0.428571
$50$	1.00000	0.141421
$51$	6.00000	0.840168
$52$	−4.00000	−0.554700
$53$	−6.00000	−0.824163	−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000	−0.824163	−0.412082	+	0.911147i	$0.635198\pi$
$54$	1.00000	0.136083
$55$	0	0
$56$	2.00000	0.267261
$57$	8.00000	1.05963
$58$	0	0
$59$	−6.00000	−0.781133	−0.390567	−	0.920575i	$-0.627721\pi$
$59$	−6.00000	−0.781133	−0.390567	+	0.920575i	$0.627721\pi$
$60$	−1.00000	−0.129099
$61$	2.00000	0.256074	0.128037	−	0.991769i	$-0.459132\pi$
$61$	2.00000	0.256074	0.128037	+	0.991769i	$0.459132\pi$
$62$	1.00000	0.127000
$63$	2.00000	0.251976
$64$	1.00000	0.125000
$65$	4.00000	0.496139
$66$	0	0
$67$	2.00000	0.244339	0.122169	−	0.992509i	$-0.461015\pi$
$67$	2.00000	0.244339	0.122169	+	0.992509i	$0.461015\pi$
$68$	6.00000	0.727607
$69$	0	0
$70$	−2.00000	−0.239046
$71$	−6.00000	−0.712069	−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000	−0.712069	−0.356034	+	0.934473i	$0.615871\pi$
$72$	1.00000	0.117851
$73$	8.00000	0.936329	0.468165	−	0.883641i	$-0.344915\pi$
$73$	8.00000	0.936329	0.468165	+	0.883641i	$0.344915\pi$
$74$	−4.00000	−0.464991
$75$	1.00000	0.115470
$76$	8.00000	0.917663
$77$	0	0
$78$	−4.00000	−0.452911
$79$	8.00000	0.900070	0.450035	−	0.893011i	$-0.351411\pi$
$79$	8.00000	0.900070	0.450035	+	0.893011i	$0.351411\pi$
$80$	−1.00000	−0.111803
$81$	1.00000	0.111111
$82$	−6.00000	−0.662589
$83$	12.0000	1.31717	0.658586	−	0.752506i	$-0.271155\pi$
$83$	12.0000	1.31717	0.658586	+	0.752506i	$0.271155\pi$
$84$	2.00000	0.218218
$85$	−6.00000	−0.650791
$86$	8.00000	0.862662
$87$	0	0
$88$	0	0
$89$	0	0		−	1.00000i	$-0.5\pi$
$89$	0	0			1.00000i	$0.5\pi$
$90$	−1.00000	−0.105409
$91$	−8.00000	−0.838628
$92$	0	0
$93$	1.00000	0.103695
$94$	−12.0000	−1.23771
$95$	−8.00000	−0.820783
$96$	1.00000	0.102062
$97$	−10.0000	−1.01535	−0.507673	−	0.861550i	$-0.669494\pi$
$97$	−10.0000	−1.01535	−0.507673	+	0.861550i	$0.669494\pi$
$98$	−3.00000	−0.303046
$99$	0	0
$100$	1.00000	0.100000
$101$	−6.00000	−0.597022	−0.298511	−	0.954406i	$-0.596490\pi$
$101$	−6.00000	−0.597022	−0.298511	+	0.954406i	$0.596490\pi$
$102$	6.00000	0.594089
$103$	−10.0000	−0.985329	−0.492665	−	0.870219i	$-0.663977\pi$
$103$	−10.0000	−0.985329	−0.492665	+	0.870219i	$0.663977\pi$
$104$	−4.00000	−0.392232
$105$	−2.00000	−0.195180
$106$	−6.00000	−0.582772
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	1.00000	0.0962250
$109$	2.00000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	−4.00000	−0.379663
$112$	2.00000	0.188982
$113$	−18.0000	−1.69330	−0.846649	−	0.532152i	$-0.821383\pi$
$113$	−18.0000	−1.69330	−0.846649	+	0.532152i	$0.821383\pi$
$114$	8.00000	0.749269
$115$	0	0
$116$	0	0
$117$	−4.00000	−0.369800
$118$	−6.00000	−0.552345
$119$	12.0000	1.10004
$120$	−1.00000	−0.0912871
$121$	−11.0000	−1.00000
$122$	2.00000	0.181071
$123$	−6.00000	−0.541002
$124$	1.00000	0.0898027
$125$	−1.00000	−0.0894427
$126$	2.00000	0.178174
$127$	−16.0000	−1.41977	−0.709885	−	0.704317i	$-0.751253\pi$
$127$	−16.0000	−1.41977	−0.709885	+	0.704317i	$0.751253\pi$
$128$	1.00000	0.0883883
$129$	8.00000	0.704361
$130$	4.00000	0.350823
$131$	−6.00000	−0.524222	−0.262111	−	0.965038i	$-0.584419\pi$
$131$	−6.00000	−0.524222	−0.262111	+	0.965038i	$0.584419\pi$
$132$	0	0
$133$	16.0000	1.38738
$134$	2.00000	0.172774
$135$	−1.00000	−0.0860663
$136$	6.00000	0.514496
$137$	−6.00000	−0.512615	−0.256307	−	0.966595i	$-0.582506\pi$
$137$	−6.00000	−0.512615	−0.256307	+	0.966595i	$0.582506\pi$
$138$	0	0
$139$	−4.00000	−0.339276	−0.169638	−	0.985506i	$-0.554260\pi$
$139$	−4.00000	−0.339276	−0.169638	+	0.985506i	$0.554260\pi$
$140$	−2.00000	−0.169031
$141$	−12.0000	−1.01058
$142$	−6.00000	−0.503509
$143$	0	0
$144$	1.00000	0.0833333
$145$	0	0
$146$	8.00000	0.662085
$147$	−3.00000	−0.247436
$148$	−4.00000	−0.328798
$149$	18.0000	1.47462	0.737309	−	0.675556i	$-0.236096\pi$
$149$	18.0000	1.47462	0.737309	+	0.675556i	$0.236096\pi$
$150$	1.00000	0.0816497
$151$	8.00000	0.651031	0.325515	−	0.945537i	$-0.394462\pi$
$151$	8.00000	0.651031	0.325515	+	0.945537i	$0.394462\pi$
$152$	8.00000	0.648886
$153$	6.00000	0.485071
$154$	0	0
$155$	−1.00000	−0.0803219
$156$	−4.00000	−0.320256
$157$	14.0000	1.11732	0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000	1.11732	0.558661	+	0.829396i	$0.311315\pi$
$158$	8.00000	0.636446
$159$	−6.00000	−0.475831
$160$	−1.00000	−0.0790569
$161$	0	0
$162$	1.00000	0.0785674
$163$	2.00000	0.156652	0.0783260	−	0.996928i	$-0.475042\pi$
$163$	2.00000	0.156652	0.0783260	+	0.996928i	$0.475042\pi$
$164$	−6.00000	−0.468521
$165$	0	0
$166$	12.0000	0.931381
$167$	−24.0000	−1.85718	−0.928588	−	0.371113i	$-0.878976\pi$
$167$	−24.0000	−1.85718	−0.928588	+	0.371113i	$0.878976\pi$
$168$	2.00000	0.154303
$169$	3.00000	0.230769
$170$	−6.00000	−0.460179
$171$	8.00000	0.611775
$172$	8.00000	0.609994
$173$	−6.00000	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−6.00000	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	2.00000	0.151186
$176$	0	0
$177$	−6.00000	−0.450988
$178$	0	0
$179$	0	0		−	1.00000i	$-0.5\pi$
$179$	0	0			1.00000i	$0.5\pi$
$180$	−1.00000	−0.0745356
$181$	2.00000	0.148659	0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000	0.148659	0.0743294	+	0.997234i	$0.476318\pi$
$182$	−8.00000	−0.592999
$183$	2.00000	0.147844
$184$	0	0
$185$	4.00000	0.294086
$186$	1.00000	0.0733236
$187$	0	0
$188$	−12.0000	−0.875190
$189$	2.00000	0.145479
$190$	−8.00000	−0.580381
$191$	−18.0000	−1.30243	−0.651217	−	0.758891i	$-0.725741\pi$
$191$	−18.0000	−1.30243	−0.651217	+	0.758891i	$0.725741\pi$
$192$	1.00000	0.0721688
$193$	14.0000	1.00774	0.503871	−	0.863779i	$-0.331909\pi$
$193$	14.0000	1.00774	0.503871	+	0.863779i	$0.331909\pi$
$194$	−10.0000	−0.717958
$195$	4.00000	0.286446
$196$	−3.00000	−0.214286
$197$	6.00000	0.427482	0.213741	−	0.976890i	$-0.431435\pi$
$197$	6.00000	0.427482	0.213741	+	0.976890i	$0.431435\pi$
$198$	0	0
$199$	8.00000	0.567105	0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000	0.567105	0.283552	+	0.958957i	$0.408487\pi$
$200$	1.00000	0.0707107
$201$	2.00000	0.141069
$202$	−6.00000	−0.422159
$203$	0	0
$204$	6.00000	0.420084
$205$	6.00000	0.419058
$206$	−10.0000	−0.696733
$207$	0	0
$208$	−4.00000	−0.277350
$209$	0	0
$210$	−2.00000	−0.138013
$211$	8.00000	0.550743	0.275371	−	0.961338i	$-0.411199\pi$
$211$	8.00000	0.550743	0.275371	+	0.961338i	$0.411199\pi$
$212$	−6.00000	−0.412082
$213$	−6.00000	−0.411113
$214$	0	0
$215$	−8.00000	−0.545595
$216$	1.00000	0.0680414
$217$	2.00000	0.135769
$218$	2.00000	0.135457
$219$	8.00000	0.540590
$220$	0	0
$221$	−24.0000	−1.61441
$222$	−4.00000	−0.268462
$223$	−28.0000	−1.87502	−0.937509	−	0.347960i	$-0.886874\pi$
$223$	−28.0000	−1.87502	−0.937509	+	0.347960i	$0.886874\pi$
$224$	2.00000	0.133631
$225$	1.00000	0.0666667
$226$	−18.0000	−1.19734
$227$	12.0000	0.796468	0.398234	−	0.917284i	$-0.369623\pi$
$227$	12.0000	0.796468	0.398234	+	0.917284i	$0.369623\pi$
$228$	8.00000	0.529813
$229$	14.0000	0.925146	0.462573	−	0.886581i	$-0.346926\pi$
$229$	14.0000	0.925146	0.462573	+	0.886581i	$0.346926\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	18.0000	1.17922	0.589610	−	0.807688i	$-0.299282\pi$
$233$	18.0000	1.17922	0.589610	+	0.807688i	$0.299282\pi$
$234$	−4.00000	−0.261488
$235$	12.0000	0.782794
$236$	−6.00000	−0.390567
$237$	8.00000	0.519656
$238$	12.0000	0.777844
$239$	12.0000	0.776215	0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000	0.776215	0.388108	+	0.921614i	$0.373129\pi$
$240$	−1.00000	−0.0645497
$241$	2.00000	0.128831	0.0644157	−	0.997923i	$-0.479482\pi$
$241$	2.00000	0.128831	0.0644157	+	0.997923i	$0.479482\pi$
$242$	−11.0000	−0.707107
$243$	1.00000	0.0641500
$244$	2.00000	0.128037
$245$	3.00000	0.191663
$246$	−6.00000	−0.382546
$247$	−32.0000	−2.03611
$248$	1.00000	0.0635001
$249$	12.0000	0.760469
$250$	−1.00000	−0.0632456
$251$	0	0		−	1.00000i	$-0.5\pi$
$251$	0	0			1.00000i	$0.5\pi$
$252$	2.00000	0.125988
$253$	0	0
$254$	−16.0000	−1.00393
$255$	−6.00000	−0.375735
$256$	1.00000	0.0625000
$257$	−30.0000	−1.87135	−0.935674	−	0.352865i	$-0.885208\pi$
$257$	−30.0000	−1.87135	−0.935674	+	0.352865i	$0.885208\pi$
$258$	8.00000	0.498058
$259$	−8.00000	−0.497096
$260$	4.00000	0.248069
$261$	0	0
$262$	−6.00000	−0.370681
$263$	0	0		−	1.00000i	$-0.5\pi$
$263$	0	0			1.00000i	$0.5\pi$
$264$	0	0
$265$	6.00000	0.368577
$266$	16.0000	0.981023
$267$	0	0
$268$	2.00000	0.122169
$269$	0	0		−	1.00000i	$-0.5\pi$
$269$	0	0			1.00000i	$0.5\pi$
$270$	−1.00000	−0.0608581
$271$	32.0000	1.94386	0.971931	−	0.235267i	$-0.0755965\pi$
$271$	32.0000	1.94386	0.971931	+	0.235267i	$0.0755965\pi$
$272$	6.00000	0.363803
$273$	−8.00000	−0.484182
$274$	−6.00000	−0.362473
$275$	0	0
$276$	0	0
$277$	8.00000	0.480673	0.240337	−	0.970690i	$-0.422742\pi$
$277$	8.00000	0.480673	0.240337	+	0.970690i	$0.422742\pi$
$278$	−4.00000	−0.239904
$279$	1.00000	0.0598684
$280$	−2.00000	−0.119523
$281$	−18.0000	−1.07379	−0.536895	−	0.843649i	$-0.680403\pi$
$281$	−18.0000	−1.07379	−0.536895	+	0.843649i	$0.680403\pi$
$282$	−12.0000	−0.714590
$283$	−22.0000	−1.30776	−0.653882	−	0.756596i	$-0.726861\pi$
$283$	−22.0000	−1.30776	−0.653882	+	0.756596i	$0.726861\pi$
$284$	−6.00000	−0.356034
$285$	−8.00000	−0.473879
$286$	0	0
$287$	−12.0000	−0.708338
$288$	1.00000	0.0589256
$289$	19.0000	1.11765
$290$	0	0
$291$	−10.0000	−0.586210
$292$	8.00000	0.468165
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	−3.00000	−0.174964
$295$	6.00000	0.349334
$296$	−4.00000	−0.232495
$297$	0	0
$298$	18.0000	1.04271
$299$	0	0
$300$	1.00000	0.0577350
$301$	16.0000	0.922225
$302$	8.00000	0.460348
$303$	−6.00000	−0.344691
$304$	8.00000	0.458831
$305$	−2.00000	−0.114520
$306$	6.00000	0.342997
$307$	−34.0000	−1.94048	−0.970241	−	0.242140i	$-0.922151\pi$
$307$	−34.0000	−1.94048	−0.970241	+	0.242140i	$0.922151\pi$
$308$	0	0
$309$	−10.0000	−0.568880
$310$	−1.00000	−0.0567962
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	−4.00000	−0.226455
$313$	−16.0000	−0.904373	−0.452187	−	0.891923i	$-0.649356\pi$
$313$	−16.0000	−0.904373	−0.452187	+	0.891923i	$0.649356\pi$
$314$	14.0000	0.790066
$315$	−2.00000	−0.112687
$316$	8.00000	0.450035
$317$	6.00000	0.336994	0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000	0.336994	0.168497	+	0.985702i	$0.446109\pi$
$318$	−6.00000	−0.336463
$319$	0	0
$320$	−1.00000	−0.0559017
$321$	0	0
$322$	0	0
$323$	48.0000	2.67079
$324$	1.00000	0.0555556
$325$	−4.00000	−0.221880
$326$	2.00000	0.110770
$327$	2.00000	0.110600
$328$	−6.00000	−0.331295
$329$	−24.0000	−1.32316
$330$	0	0
$331$	−4.00000	−0.219860	−0.109930	−	0.993939i	$-0.535063\pi$
$331$	−4.00000	−0.219860	−0.109930	+	0.993939i	$0.535063\pi$
$332$	12.0000	0.658586
$333$	−4.00000	−0.219199
$334$	−24.0000	−1.31322
$335$	−2.00000	−0.109272
$336$	2.00000	0.109109
$337$	20.0000	1.08947	0.544735	−	0.838608i	$-0.316630\pi$
$337$	20.0000	1.08947	0.544735	+	0.838608i	$0.316630\pi$
$338$	3.00000	0.163178
$339$	−18.0000	−0.977626
$340$	−6.00000	−0.325396
$341$	0	0
$342$	8.00000	0.432590
$343$	−20.0000	−1.07990
$344$	8.00000	0.431331
$345$	0	0
$346$	−6.00000	−0.322562
$347$	−36.0000	−1.93258	−0.966291	−	0.257454i	$-0.917117\pi$
$347$	−36.0000	−1.93258	−0.966291	+	0.257454i	$0.917117\pi$
$348$	0	0
$349$	2.00000	0.107058	0.0535288	−	0.998566i	$-0.482953\pi$
$349$	2.00000	0.107058	0.0535288	+	0.998566i	$0.482953\pi$
$350$	2.00000	0.106904
$351$	−4.00000	−0.213504
$352$	0	0
$353$	30.0000	1.59674	0.798369	−	0.602168i	$-0.205696\pi$
$353$	30.0000	1.59674	0.798369	+	0.602168i	$0.205696\pi$
$354$	−6.00000	−0.318896
$355$	6.00000	0.318447
$356$	0	0
$357$	12.0000	0.635107
$358$	0	0
$359$	30.0000	1.58334	0.791670	−	0.610949i	$-0.209212\pi$
$359$	30.0000	1.58334	0.791670	+	0.610949i	$0.209212\pi$
$360$	−1.00000	−0.0527046
$361$	45.0000	2.36842
$362$	2.00000	0.105118
$363$	−11.0000	−0.577350
$364$	−8.00000	−0.419314
$365$	−8.00000	−0.418739
$366$	2.00000	0.104542
$367$	20.0000	1.04399	0.521996	−	0.852948i	$-0.325188\pi$
$367$	20.0000	1.04399	0.521996	+	0.852948i	$0.325188\pi$
$368$	0	0
$369$	−6.00000	−0.312348
$370$	4.00000	0.207950
$371$	−12.0000	−0.623009
$372$	1.00000	0.0518476
$373$	38.0000	1.96757	0.983783	−	0.179364i	$-0.0574041\pi$
$373$	38.0000	1.96757	0.983783	+	0.179364i	$0.0574041\pi$
$374$	0	0
$375$	−1.00000	−0.0516398
$376$	−12.0000	−0.618853
$377$	0	0
$378$	2.00000	0.102869
$379$	32.0000	1.64373	0.821865	−	0.569683i	$-0.192934\pi$
$379$	32.0000	1.64373	0.821865	+	0.569683i	$0.192934\pi$
$380$	−8.00000	−0.410391
$381$	−16.0000	−0.819705
$382$	−18.0000	−0.920960
$383$	0	0		−	1.00000i	$-0.5\pi$
$383$	0	0			1.00000i	$0.5\pi$
$384$	1.00000	0.0510310
$385$	0	0
$386$	14.0000	0.712581
$387$	8.00000	0.406663
$388$	−10.0000	−0.507673
$389$	24.0000	1.21685	0.608424	−	0.793612i	$-0.291802\pi$
$389$	24.0000	1.21685	0.608424	+	0.793612i	$0.291802\pi$
$390$	4.00000	0.202548
$391$	0	0
$392$	−3.00000	−0.151523
$393$	−6.00000	−0.302660
$394$	6.00000	0.302276
$395$	−8.00000	−0.402524
$396$	0	0
$397$	2.00000	0.100377	0.0501886	−	0.998740i	$-0.484018\pi$
$397$	2.00000	0.100377	0.0501886	+	0.998740i	$0.484018\pi$
$398$	8.00000	0.401004
$399$	16.0000	0.801002
$400$	1.00000	0.0500000
$401$	36.0000	1.79775	0.898877	−	0.438201i	$-0.144384\pi$
$401$	36.0000	1.79775	0.898877	+	0.438201i	$0.144384\pi$
$402$	2.00000	0.0997509
$403$	−4.00000	−0.199254
$404$	−6.00000	−0.298511
$405$	−1.00000	−0.0496904
$406$	0	0
$407$	0	0
$408$	6.00000	0.297044
$409$	14.0000	0.692255	0.346128	−	0.938187i	$-0.387496\pi$
$409$	14.0000	0.692255	0.346128	+	0.938187i	$0.387496\pi$
$410$	6.00000	0.296319
$411$	−6.00000	−0.295958
$412$	−10.0000	−0.492665
$413$	−12.0000	−0.590481
$414$	0	0
$415$	−12.0000	−0.589057
$416$	−4.00000	−0.196116
$417$	−4.00000	−0.195881
$418$	0	0
$419$	30.0000	1.46560	0.732798	−	0.680446i	$-0.238214\pi$
$419$	30.0000	1.46560	0.732798	+	0.680446i	$0.238214\pi$
$420$	−2.00000	−0.0975900
$421$	26.0000	1.26716	0.633581	−	0.773676i	$-0.281584\pi$
$421$	26.0000	1.26716	0.633581	+	0.773676i	$0.281584\pi$
$422$	8.00000	0.389434
$423$	−12.0000	−0.583460
$424$	−6.00000	−0.291386
$425$	6.00000	0.291043
$426$	−6.00000	−0.290701
$427$	4.00000	0.193574
$428$	0	0
$429$	0	0
$430$	−8.00000	−0.385794
$431$	6.00000	0.289010	0.144505	−	0.989504i	$-0.453841\pi$
$431$	6.00000	0.289010	0.144505	+	0.989504i	$0.453841\pi$
$432$	1.00000	0.0481125
$433$	−16.0000	−0.768911	−0.384455	−	0.923144i	$-0.625611\pi$
$433$	−16.0000	−0.768911	−0.384455	+	0.923144i	$0.625611\pi$
$434$	2.00000	0.0960031
$435$	0	0
$436$	2.00000	0.0957826
$437$	0	0
$438$	8.00000	0.382255
$439$	8.00000	0.381819	0.190910	−	0.981608i	$-0.438856\pi$
$439$	8.00000	0.381819	0.190910	+	0.981608i	$0.438856\pi$
$440$	0	0
$441$	−3.00000	−0.142857
$442$	−24.0000	−1.14156
$443$	−36.0000	−1.71041	−0.855206	−	0.518289i	$-0.826569\pi$
$443$	−36.0000	−1.71041	−0.855206	+	0.518289i	$0.826569\pi$
$444$	−4.00000	−0.189832
$445$	0	0
$446$	−28.0000	−1.32584
$447$	18.0000	0.851371
$448$	2.00000	0.0944911
$449$	12.0000	0.566315	0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000	0.566315	0.283158	+	0.959073i	$0.408618\pi$
$450$	1.00000	0.0471405
$451$	0	0
$452$	−18.0000	−0.846649
$453$	8.00000	0.375873
$454$	12.0000	0.563188
$455$	8.00000	0.375046
$456$	8.00000	0.374634
$457$	8.00000	0.374224	0.187112	−	0.982339i	$-0.440087\pi$
$457$	8.00000	0.374224	0.187112	+	0.982339i	$0.440087\pi$
$458$	14.0000	0.654177
$459$	6.00000	0.280056
$460$	0	0
$461$	−36.0000	−1.67669	−0.838344	−	0.545142i	$-0.816476\pi$
$461$	−36.0000	−1.67669	−0.838344	+	0.545142i	$0.816476\pi$
$462$	0	0
$463$	−16.0000	−0.743583	−0.371792	−	0.928316i	$-0.621256\pi$
$463$	−16.0000	−0.743583	−0.371792	+	0.928316i	$0.621256\pi$
$464$	0	0
$465$	−1.00000	−0.0463739
$466$	18.0000	0.833834
$467$	24.0000	1.11059	0.555294	−	0.831654i	$-0.312606\pi$
$467$	24.0000	1.11059	0.555294	+	0.831654i	$0.312606\pi$
$468$	−4.00000	−0.184900
$469$	4.00000	0.184703
$470$	12.0000	0.553519
$471$	14.0000	0.645086
$472$	−6.00000	−0.276172
$473$	0	0
$474$	8.00000	0.367452
$475$	8.00000	0.367065
$476$	12.0000	0.550019
$477$	−6.00000	−0.274721
$478$	12.0000	0.548867
$479$	−6.00000	−0.274147	−0.137073	−	0.990561i	$-0.543770\pi$
$479$	−6.00000	−0.274147	−0.137073	+	0.990561i	$0.543770\pi$
$480$	−1.00000	−0.0456435
$481$	16.0000	0.729537
$482$	2.00000	0.0910975
$483$	0	0
$484$	−11.0000	−0.500000
$485$	10.0000	0.454077
$486$	1.00000	0.0453609
$487$	−40.0000	−1.81257	−0.906287	−	0.422664i	$-0.861095\pi$
$487$	−40.0000	−1.81257	−0.906287	+	0.422664i	$0.861095\pi$
$488$	2.00000	0.0905357
$489$	2.00000	0.0904431
$490$	3.00000	0.135526
$491$	12.0000	0.541552	0.270776	−	0.962642i	$-0.412720\pi$
$491$	12.0000	0.541552	0.270776	+	0.962642i	$0.412720\pi$
$492$	−6.00000	−0.270501
$493$	0	0
$494$	−32.0000	−1.43975
$495$	0	0
$496$	1.00000	0.0449013
$497$	−12.0000	−0.538274
$498$	12.0000	0.537733
$499$	−4.00000	−0.179065	−0.0895323	−	0.995984i	$-0.528537\pi$
$499$	−4.00000	−0.179065	−0.0895323	+	0.995984i	$0.528537\pi$
$500$	−1.00000	−0.0447214
$501$	−24.0000	−1.07224
$502$	0	0
$503$	36.0000	1.60516	0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000	1.60516	0.802580	+	0.596544i	$0.203460\pi$
$504$	2.00000	0.0890871
$505$	6.00000	0.266996
$506$	0	0
$507$	3.00000	0.133235
$508$	−16.0000	−0.709885
$509$	12.0000	0.531891	0.265945	−	0.963988i	$-0.414316\pi$
$509$	12.0000	0.531891	0.265945	+	0.963988i	$0.414316\pi$
$510$	−6.00000	−0.265684
$511$	16.0000	0.707798
$512$	1.00000	0.0441942
$513$	8.00000	0.353209
$514$	−30.0000	−1.32324
$515$	10.0000	0.440653
$516$	8.00000	0.352180
$517$	0	0
$518$	−8.00000	−0.351500
$519$	−6.00000	−0.263371
$520$	4.00000	0.175412
$521$	−6.00000	−0.262865	−0.131432	−	0.991325i	$-0.541958\pi$
$521$	−6.00000	−0.262865	−0.131432	+	0.991325i	$0.541958\pi$
$522$	0	0
$523$	−40.0000	−1.74908	−0.874539	−	0.484955i	$-0.838836\pi$
$523$	−40.0000	−1.74908	−0.874539	+	0.484955i	$0.838836\pi$
$524$	−6.00000	−0.262111
$525$	2.00000	0.0872872
$526$	0	0
$527$	6.00000	0.261364
$528$	0	0
$529$	−23.0000	−1.00000
$530$	6.00000	0.260623
$531$	−6.00000	−0.260378
$532$	16.0000	0.693688
$533$	24.0000	1.03956
$534$	0	0
$535$	0	0
$536$	2.00000	0.0863868
$537$	0	0
$538$	0	0
$539$	0	0
$540$	−1.00000	−0.0430331
$541$	2.00000	0.0859867	0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000	0.0859867	0.0429934	+	0.999075i	$0.486311\pi$
$542$	32.0000	1.37452
$543$	2.00000	0.0858282
$544$	6.00000	0.257248
$545$	−2.00000	−0.0856706
$546$	−8.00000	−0.342368
$547$	2.00000	0.0855138	0.0427569	−	0.999086i	$-0.486386\pi$
$547$	2.00000	0.0855138	0.0427569	+	0.999086i	$0.486386\pi$
$548$	−6.00000	−0.256307
$549$	2.00000	0.0853579
$550$	0	0
$551$	0	0
$552$	0	0
$553$	16.0000	0.680389
$554$	8.00000	0.339887
$555$	4.00000	0.169791
$556$	−4.00000	−0.169638
$557$	42.0000	1.77960	0.889799	−	0.456354i	$-0.150845\pi$
$557$	42.0000	1.77960	0.889799	+	0.456354i	$0.150845\pi$
$558$	1.00000	0.0423334
$559$	−32.0000	−1.35346
$560$	−2.00000	−0.0845154
$561$	0	0
$562$	−18.0000	−0.759284
$563$	−24.0000	−1.01148	−0.505740	−	0.862686i	$-0.668780\pi$
$563$	−24.0000	−1.01148	−0.505740	+	0.862686i	$0.668780\pi$
$564$	−12.0000	−0.505291
$565$	18.0000	0.757266
$566$	−22.0000	−0.924729
$567$	2.00000	0.0839921
$568$	−6.00000	−0.251754
$569$	36.0000	1.50920	0.754599	−	0.656186i	$-0.227831\pi$
$569$	36.0000	1.50920	0.754599	+	0.656186i	$0.227831\pi$
$570$	−8.00000	−0.335083
$571$	20.0000	0.836974	0.418487	−	0.908223i	$-0.362561\pi$
$571$	20.0000	0.836974	0.418487	+	0.908223i	$0.362561\pi$
$572$	0	0
$573$	−18.0000	−0.751961
$574$	−12.0000	−0.500870
$575$	0	0
$576$	1.00000	0.0416667
$577$	−22.0000	−0.915872	−0.457936	−	0.888985i	$-0.651411\pi$
$577$	−22.0000	−0.915872	−0.457936	+	0.888985i	$0.651411\pi$
$578$	19.0000	0.790296
$579$	14.0000	0.581820
$580$	0	0
$581$	24.0000	0.995688
$582$	−10.0000	−0.414513
$583$	0	0
$584$	8.00000	0.331042
$585$	4.00000	0.165380
$586$	6.00000	0.247858
$587$	−12.0000	−0.495293	−0.247647	−	0.968850i	$-0.579657\pi$
$587$	−12.0000	−0.495293	−0.247647	+	0.968850i	$0.579657\pi$
$588$	−3.00000	−0.123718
$589$	8.00000	0.329634
$590$	6.00000	0.247016
$591$	6.00000	0.246807
$592$	−4.00000	−0.164399
$593$	6.00000	0.246390	0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000	0.246390	0.123195	+	0.992382i	$0.460686\pi$
$594$	0	0
$595$	−12.0000	−0.491952
$596$	18.0000	0.737309
$597$	8.00000	0.327418
$598$	0	0
$599$	6.00000	0.245153	0.122577	−	0.992459i	$-0.460884\pi$
$599$	6.00000	0.245153	0.122577	+	0.992459i	$0.460884\pi$
$600$	1.00000	0.0408248
$601$	−10.0000	−0.407909	−0.203954	−	0.978980i	$-0.565379\pi$
$601$	−10.0000	−0.407909	−0.203954	+	0.978980i	$0.565379\pi$
$602$	16.0000	0.652111
$603$	2.00000	0.0814463
$604$	8.00000	0.325515
$605$	11.0000	0.447214
$606$	−6.00000	−0.243733
$607$	−34.0000	−1.38002	−0.690009	−	0.723801i	$-0.742393\pi$
$607$	−34.0000	−1.38002	−0.690009	+	0.723801i	$0.742393\pi$
$608$	8.00000	0.324443
$609$	0	0
$610$	−2.00000	−0.0809776
$611$	48.0000	1.94187
$612$	6.00000	0.242536
$613$	−40.0000	−1.61558	−0.807792	−	0.589467i	$-0.799338\pi$
$613$	−40.0000	−1.61558	−0.807792	+	0.589467i	$0.799338\pi$
$614$	−34.0000	−1.37213
$615$	6.00000	0.241943
$616$	0	0
$617$	6.00000	0.241551	0.120775	−	0.992680i	$-0.461462\pi$
$617$	6.00000	0.241551	0.120775	+	0.992680i	$0.461462\pi$
$618$	−10.0000	−0.402259
$619$	−4.00000	−0.160774	−0.0803868	−	0.996764i	$-0.525616\pi$
$619$	−4.00000	−0.160774	−0.0803868	+	0.996764i	$0.525616\pi$
$620$	−1.00000	−0.0401610
$621$	0	0
$622$	−18.0000	−0.721734
$623$	0	0
$624$	−4.00000	−0.160128
$625$	1.00000	0.0400000
$626$	−16.0000	−0.639489
$627$	0	0
$628$	14.0000	0.558661
$629$	−24.0000	−0.956943
$630$	−2.00000	−0.0796819
$631$	8.00000	0.318475	0.159237	−	0.987240i	$-0.449096\pi$
$631$	8.00000	0.318475	0.159237	+	0.987240i	$0.449096\pi$
$632$	8.00000	0.318223
$633$	8.00000	0.317971
$634$	6.00000	0.238290
$635$	16.0000	0.634941
$636$	−6.00000	−0.237915
$637$	12.0000	0.475457
$638$	0	0
$639$	−6.00000	−0.237356
$640$	−1.00000	−0.0395285
$641$	−12.0000	−0.473972	−0.236986	−	0.971513i	$-0.576159\pi$
$641$	−12.0000	−0.473972	−0.236986	+	0.971513i	$0.576159\pi$
$642$	0	0
$643$	32.0000	1.26196	0.630978	−	0.775800i	$-0.282654\pi$
$643$	32.0000	1.26196	0.630978	+	0.775800i	$0.282654\pi$
$644$	0	0
$645$	−8.00000	−0.315000
$646$	48.0000	1.88853
$647$	0	0		−	1.00000i	$-0.5\pi$
$647$	0	0			1.00000i	$0.5\pi$
$648$	1.00000	0.0392837
$649$	0	0
$650$	−4.00000	−0.156893
$651$	2.00000	0.0783862
$652$	2.00000	0.0783260
$653$	6.00000	0.234798	0.117399	−	0.993085i	$-0.462544\pi$
$653$	6.00000	0.234798	0.117399	+	0.993085i	$0.462544\pi$
$654$	2.00000	0.0782062
$655$	6.00000	0.234439
$656$	−6.00000	−0.234261
$657$	8.00000	0.312110
$658$	−24.0000	−0.935617
$659$	6.00000	0.233727	0.116863	−	0.993148i	$-0.462716\pi$
$659$	6.00000	0.233727	0.116863	+	0.993148i	$0.462716\pi$
$660$	0	0
$661$	−10.0000	−0.388955	−0.194477	−	0.980907i	$-0.562301\pi$
$661$	−10.0000	−0.388955	−0.194477	+	0.980907i	$0.562301\pi$
$662$	−4.00000	−0.155464
$663$	−24.0000	−0.932083
$664$	12.0000	0.465690
$665$	−16.0000	−0.620453
$666$	−4.00000	−0.154997
$667$	0	0
$668$	−24.0000	−0.928588
$669$	−28.0000	−1.08254
$670$	−2.00000	−0.0772667
$671$	0	0
$672$	2.00000	0.0771517
$673$	44.0000	1.69608	0.848038	−	0.529936i	$-0.177784\pi$
$673$	44.0000	1.69608	0.848038	+	0.529936i	$0.177784\pi$
$674$	20.0000	0.770371
$675$	1.00000	0.0384900
$676$	3.00000	0.115385
$677$	18.0000	0.691796	0.345898	−	0.938272i	$-0.387574\pi$
$677$	18.0000	0.691796	0.345898	+	0.938272i	$0.387574\pi$
$678$	−18.0000	−0.691286
$679$	−20.0000	−0.767530
$680$	−6.00000	−0.230089
$681$	12.0000	0.459841
$682$	0	0
$683$	−36.0000	−1.37750	−0.688751	−	0.724998i	$-0.741841\pi$
$683$	−36.0000	−1.37750	−0.688751	+	0.724998i	$0.741841\pi$
$684$	8.00000	0.305888
$685$	6.00000	0.229248
$686$	−20.0000	−0.763604
$687$	14.0000	0.534133
$688$	8.00000	0.304997
$689$	24.0000	0.914327
$690$	0	0
$691$	44.0000	1.67384	0.836919	−	0.547326i	$-0.184354\pi$
$691$	44.0000	1.67384	0.836919	+	0.547326i	$0.184354\pi$
$692$	−6.00000	−0.228086
$693$	0	0
$694$	−36.0000	−1.36654
$695$	4.00000	0.151729
$696$	0	0
$697$	−36.0000	−1.36360
$698$	2.00000	0.0757011
$699$	18.0000	0.680823
$700$	2.00000	0.0755929
$701$	30.0000	1.13308	0.566542	−	0.824033i	$-0.308281\pi$
$701$	30.0000	1.13308	0.566542	+	0.824033i	$0.308281\pi$
$702$	−4.00000	−0.150970
$703$	−32.0000	−1.20690
$704$	0	0
$705$	12.0000	0.451946
$706$	30.0000	1.12906
$707$	−12.0000	−0.451306
$708$	−6.00000	−0.225494
$709$	−10.0000	−0.375558	−0.187779	−	0.982211i	$-0.560129\pi$
$709$	−10.0000	−0.375558	−0.187779	+	0.982211i	$0.560129\pi$
$710$	6.00000	0.225176
$711$	8.00000	0.300023
$712$	0	0
$713$	0	0
$714$	12.0000	0.449089
$715$	0	0
$716$	0	0
$717$	12.0000	0.448148
$718$	30.0000	1.11959
$719$	−24.0000	−0.895049	−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000	−0.895049	−0.447524	+	0.894272i	$0.647694\pi$
$720$	−1.00000	−0.0372678
$721$	−20.0000	−0.744839
$722$	45.0000	1.67473
$723$	2.00000	0.0743808
$724$	2.00000	0.0743294
$725$	0	0
$726$	−11.0000	−0.408248
$727$	26.0000	0.964287	0.482143	−	0.876092i	$-0.339858\pi$
$727$	26.0000	0.964287	0.482143	+	0.876092i	$0.339858\pi$
$728$	−8.00000	−0.296500
$729$	1.00000	0.0370370
$730$	−8.00000	−0.296093
$731$	48.0000	1.77534
$732$	2.00000	0.0739221
$733$	−22.0000	−0.812589	−0.406294	−	0.913742i	$-0.633179\pi$
$733$	−22.0000	−0.812589	−0.406294	+	0.913742i	$0.633179\pi$
$734$	20.0000	0.738213
$735$	3.00000	0.110657
$736$	0	0
$737$	0	0
$738$	−6.00000	−0.220863
$739$	−52.0000	−1.91285	−0.956425	−	0.291977i	$-0.905687\pi$
$739$	−52.0000	−1.91285	−0.956425	+	0.291977i	$0.905687\pi$
$740$	4.00000	0.147043
$741$	−32.0000	−1.17555
$742$	−12.0000	−0.440534
$743$	24.0000	0.880475	0.440237	−	0.897881i	$-0.354894\pi$
$743$	24.0000	0.880475	0.440237	+	0.897881i	$0.354894\pi$
$744$	1.00000	0.0366618
$745$	−18.0000	−0.659469
$746$	38.0000	1.39128
$747$	12.0000	0.439057
$748$	0	0
$749$	0	0
$750$	−1.00000	−0.0365148
$751$	−16.0000	−0.583848	−0.291924	−	0.956441i	$-0.594295\pi$
$751$	−16.0000	−0.583848	−0.291924	+	0.956441i	$0.594295\pi$
$752$	−12.0000	−0.437595
$753$	0	0
$754$	0	0
$755$	−8.00000	−0.291150
$756$	2.00000	0.0727393
$757$	−4.00000	−0.145382	−0.0726912	−	0.997354i	$-0.523159\pi$
$757$	−4.00000	−0.145382	−0.0726912	+	0.997354i	$0.523159\pi$
$758$	32.0000	1.16229
$759$	0	0
$760$	−8.00000	−0.290191
$761$	12.0000	0.435000	0.217500	−	0.976060i	$-0.430210\pi$
$761$	12.0000	0.435000	0.217500	+	0.976060i	$0.430210\pi$
$762$	−16.0000	−0.579619
$763$	4.00000	0.144810
$764$	−18.0000	−0.651217
$765$	−6.00000	−0.216930
$766$	0	0
$767$	24.0000	0.866590
$768$	1.00000	0.0360844
$769$	−34.0000	−1.22607	−0.613036	−	0.790055i	$-0.710052\pi$
$769$	−34.0000	−1.22607	−0.613036	+	0.790055i	$0.710052\pi$
$770$	0	0
$771$	−30.0000	−1.08042
$772$	14.0000	0.503871
$773$	−54.0000	−1.94225	−0.971123	−	0.238581i	$-0.923318\pi$
$773$	−54.0000	−1.94225	−0.971123	+	0.238581i	$0.923318\pi$
$774$	8.00000	0.287554
$775$	1.00000	0.0359211
$776$	−10.0000	−0.358979
$777$	−8.00000	−0.286998
$778$	24.0000	0.860442
$779$	−48.0000	−1.71978
$780$	4.00000	0.143223
$781$	0	0
$782$	0	0
$783$	0	0
$784$	−3.00000	−0.107143
$785$	−14.0000	−0.499681
$786$	−6.00000	−0.214013
$787$	20.0000	0.712923	0.356462	−	0.934310i	$-0.383983\pi$
$787$	20.0000	0.712923	0.356462	+	0.934310i	$0.383983\pi$
$788$	6.00000	0.213741
$789$	0	0
$790$	−8.00000	−0.284627
$791$	−36.0000	−1.28001
$792$	0	0
$793$	−8.00000	−0.284088
$794$	2.00000	0.0709773
$795$	6.00000	0.212798
$796$	8.00000	0.283552
$797$	6.00000	0.212531	0.106265	−	0.994338i	$-0.466111\pi$
$797$	6.00000	0.212531	0.106265	+	0.994338i	$0.466111\pi$
$798$	16.0000	0.566394
$799$	−72.0000	−2.54718
$800$	1.00000	0.0353553
$801$	0	0
$802$	36.0000	1.27120
$803$	0	0
$804$	2.00000	0.0705346
$805$	0	0
$806$	−4.00000	−0.140894
$807$	0	0
$808$	−6.00000	−0.211079
$809$	36.0000	1.26569	0.632846	−	0.774277i	$-0.281886\pi$
$809$	36.0000	1.26569	0.632846	+	0.774277i	$0.281886\pi$
$810$	−1.00000	−0.0351364
$811$	−16.0000	−0.561836	−0.280918	−	0.959732i	$-0.590639\pi$
$811$	−16.0000	−0.561836	−0.280918	+	0.959732i	$0.590639\pi$
$812$	0	0
$813$	32.0000	1.12229
$814$	0	0
$815$	−2.00000	−0.0700569
$816$	6.00000	0.210042
$817$	64.0000	2.23908
$818$	14.0000	0.489499
$819$	−8.00000	−0.279543
$820$	6.00000	0.209529
$821$	−24.0000	−0.837606	−0.418803	−	0.908077i	$-0.637550\pi$
$821$	−24.0000	−0.837606	−0.418803	+	0.908077i	$0.637550\pi$
$822$	−6.00000	−0.209274
$823$	−16.0000	−0.557725	−0.278862	−	0.960331i	$-0.589957\pi$
$823$	−16.0000	−0.557725	−0.278862	+	0.960331i	$0.589957\pi$
$824$	−10.0000	−0.348367
$825$	0	0
$826$	−12.0000	−0.417533
$827$	−12.0000	−0.417281	−0.208640	−	0.977992i	$-0.566904\pi$
$827$	−12.0000	−0.417281	−0.208640	+	0.977992i	$0.566904\pi$
$828$	0	0
$829$	−22.0000	−0.764092	−0.382046	−	0.924143i	$-0.624780\pi$
$829$	−22.0000	−0.764092	−0.382046	+	0.924143i	$0.624780\pi$
$830$	−12.0000	−0.416526
$831$	8.00000	0.277517
$832$	−4.00000	−0.138675
$833$	−18.0000	−0.623663
$834$	−4.00000	−0.138509
$835$	24.0000	0.830554
$836$	0	0
$837$	1.00000	0.0345651
$838$	30.0000	1.03633
$839$	−18.0000	−0.621429	−0.310715	−	0.950503i	$-0.600568\pi$
$839$	−18.0000	−0.621429	−0.310715	+	0.950503i	$0.600568\pi$
$840$	−2.00000	−0.0690066
$841$	−29.0000	−1.00000
$842$	26.0000	0.896019
$843$	−18.0000	−0.619953
$844$	8.00000	0.275371
$845$	−3.00000	−0.103203
$846$	−12.0000	−0.412568
$847$	−22.0000	−0.755929
$848$	−6.00000	−0.206041
$849$	−22.0000	−0.755038
$850$	6.00000	0.205798
$851$	0	0
$852$	−6.00000	−0.205557
$853$	2.00000	0.0684787	0.0342393	−	0.999414i	$-0.489099\pi$
$853$	2.00000	0.0684787	0.0342393	+	0.999414i	$0.489099\pi$
$854$	4.00000	0.136877
$855$	−8.00000	−0.273594
$856$	0	0
$857$	54.0000	1.84460	0.922302	−	0.386469i	$-0.126305\pi$
$857$	54.0000	1.84460	0.922302	+	0.386469i	$0.126305\pi$
$858$	0	0
$859$	20.0000	0.682391	0.341196	−	0.939992i	$-0.389168\pi$
$859$	20.0000	0.682391	0.341196	+	0.939992i	$0.389168\pi$
$860$	−8.00000	−0.272798
$861$	−12.0000	−0.408959
$862$	6.00000	0.204361
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	1.00000	0.0340207
$865$	6.00000	0.204006
$866$	−16.0000	−0.543702
$867$	19.0000	0.645274
$868$	2.00000	0.0678844
$869$	0	0
$870$	0	0
$871$	−8.00000	−0.271070
$872$	2.00000	0.0677285
$873$	−10.0000	−0.338449
$874$	0	0
$875$	−2.00000	−0.0676123
$876$	8.00000	0.270295
$877$	−22.0000	−0.742887	−0.371444	−	0.928456i	$-0.621137\pi$
$877$	−22.0000	−0.742887	−0.371444	+	0.928456i	$0.621137\pi$
$878$	8.00000	0.269987
$879$	6.00000	0.202375
$880$	0	0
$881$	0	0		−	1.00000i	$-0.5\pi$
$881$	0	0			1.00000i	$0.5\pi$
$882$	−3.00000	−0.101015
$883$	44.0000	1.48072	0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000	1.48072	0.740359	+	0.672212i	$0.234656\pi$
$884$	−24.0000	−0.807207
$885$	6.00000	0.201688
$886$	−36.0000	−1.20944
$887$	24.0000	0.805841	0.402921	−	0.915235i	$-0.367995\pi$
$887$	24.0000	0.805841	0.402921	+	0.915235i	$0.367995\pi$
$888$	−4.00000	−0.134231
$889$	−32.0000	−1.07325
$890$	0	0
$891$	0	0
$892$	−28.0000	−0.937509
$893$	−96.0000	−3.21252
$894$	18.0000	0.602010
$895$	0	0
$896$	2.00000	0.0668153
$897$	0	0
$898$	12.0000	0.400445
$899$	0	0
$900$	1.00000	0.0333333
$901$	−36.0000	−1.19933
$902$	0	0
$903$	16.0000	0.532447
$904$	−18.0000	−0.598671
$905$	−2.00000	−0.0664822
$906$	8.00000	0.265782
$907$	−10.0000	−0.332045	−0.166022	−	0.986122i	$-0.553092\pi$
$907$	−10.0000	−0.332045	−0.166022	+	0.986122i	$0.553092\pi$
$908$	12.0000	0.398234
$909$	−6.00000	−0.199007
$910$	8.00000	0.265197
$911$	24.0000	0.795155	0.397578	−	0.917568i	$-0.369851\pi$
$911$	24.0000	0.795155	0.397578	+	0.917568i	$0.369851\pi$
$912$	8.00000	0.264906
$913$	0	0
$914$	8.00000	0.264616
$915$	−2.00000	−0.0661180
$916$	14.0000	0.462573
$917$	−12.0000	−0.396275
$918$	6.00000	0.198030
$919$	−52.0000	−1.71532	−0.857661	−	0.514216i	$-0.828083\pi$
$919$	−52.0000	−1.71532	−0.857661	+	0.514216i	$0.828083\pi$
$920$	0	0
$921$	−34.0000	−1.12034
$922$	−36.0000	−1.18560
$923$	24.0000	0.789970
$924$	0	0
$925$	−4.00000	−0.131519
$926$	−16.0000	−0.525793
$927$	−10.0000	−0.328443
$928$	0	0
$929$	0	0		−	1.00000i	$-0.5\pi$
$929$	0	0			1.00000i	$0.5\pi$
$930$	−1.00000	−0.0327913
$931$	−24.0000	−0.786568
$932$	18.0000	0.589610
$933$	−18.0000	−0.589294
$934$	24.0000	0.785304
$935$	0	0
$936$	−4.00000	−0.130744
$937$	38.0000	1.24141	0.620703	−	0.784046i	$-0.286847\pi$
$937$	38.0000	1.24141	0.620703	+	0.784046i	$0.286847\pi$
$938$	4.00000	0.130605
$939$	−16.0000	−0.522140
$940$	12.0000	0.391397
$941$	−48.0000	−1.56476	−0.782378	−	0.622804i	$-0.785993\pi$
$941$	−48.0000	−1.56476	−0.782378	+	0.622804i	$0.785993\pi$
$942$	14.0000	0.456145
$943$	0	0
$944$	−6.00000	−0.195283
$945$	−2.00000	−0.0650600
$946$	0	0
$947$	12.0000	0.389948	0.194974	−	0.980808i	$-0.437538\pi$
$947$	12.0000	0.389948	0.194974	+	0.980808i	$0.437538\pi$
$948$	8.00000	0.259828
$949$	−32.0000	−1.03876
$950$	8.00000	0.259554
$951$	6.00000	0.194563
$952$	12.0000	0.388922
$953$	18.0000	0.583077	0.291539	−	0.956559i	$-0.405833\pi$
$953$	18.0000	0.583077	0.291539	+	0.956559i	$0.405833\pi$
$954$	−6.00000	−0.194257
$955$	18.0000	0.582466
$956$	12.0000	0.388108
$957$	0	0
$958$	−6.00000	−0.193851
$959$	−12.0000	−0.387500
$960$	−1.00000	−0.0322749
$961$	1.00000	0.0322581
$962$	16.0000	0.515861
$963$	0	0
$964$	2.00000	0.0644157
$965$	−14.0000	−0.450676
$966$	0	0
$967$	−4.00000	−0.128631	−0.0643157	−	0.997930i	$-0.520486\pi$
$967$	−4.00000	−0.128631	−0.0643157	+	0.997930i	$0.520486\pi$
$968$	−11.0000	−0.353553
$969$	48.0000	1.54198
$970$	10.0000	0.321081
$971$	6.00000	0.192549	0.0962746	−	0.995355i	$-0.469307\pi$
$971$	6.00000	0.192549	0.0962746	+	0.995355i	$0.469307\pi$
$972$	1.00000	0.0320750
$973$	−8.00000	−0.256468
$974$	−40.0000	−1.28168
$975$	−4.00000	−0.128103
$976$	2.00000	0.0640184
$977$	−18.0000	−0.575871	−0.287936	−	0.957650i	$-0.592969\pi$
$977$	−18.0000	−0.575871	−0.287936	+	0.957650i	$0.592969\pi$
$978$	2.00000	0.0639529
$979$	0	0
$980$	3.00000	0.0958315
$981$	2.00000	0.0638551
$982$	12.0000	0.382935
$983$	24.0000	0.765481	0.382741	−	0.923856i	$-0.374980\pi$
$983$	24.0000	0.765481	0.382741	+	0.923856i	$0.374980\pi$
$984$	−6.00000	−0.191273
$985$	−6.00000	−0.191176
$986$	0	0
$987$	−24.0000	−0.763928
$988$	−32.0000	−1.01806
$989$	0	0
$990$	0	0
$991$	−16.0000	−0.508257	−0.254128	−	0.967170i	$-0.581789\pi$
$991$	−16.0000	−0.508257	−0.254128	+	0.967170i	$0.581789\pi$
$992$	1.00000	0.0317500
$993$	−4.00000	−0.126936
$994$	−12.0000	−0.380617
$995$	−8.00000	−0.253617
$996$	12.0000	0.380235
$997$	−10.0000	−0.316703	−0.158352	−	0.987383i	$-0.550618\pi$
$997$	−10.0000	−0.316703	−0.158352	+	0.987383i	$0.550618\pi$
$998$	−4.00000	−0.126618
$999$	−4.00000	−0.126554

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	930.2.a.n.1.1	✓	1
3.2	odd	2		2790.2.a.k.1.1		1
4.3	odd	2		7440.2.a.b.1.1		1
5.2	odd	4		4650.2.d.u.3349.2		2
5.3	odd	4		4650.2.d.u.3349.1		2
5.4	even	2		4650.2.a.d.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
930.2.a.n.1.1	✓	1	1.1	even	1	trivial
2790.2.a.k.1.1		1	3.2	odd	2
4650.2.a.d.1.1		1	5.4	even	2
4650.2.d.u.3349.1		2	5.3	odd	4
4650.2.d.u.3349.2		2	5.2	odd	4
7440.2.a.b.1.1		1	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 930 = 2 \cdot 3 \cdot 5 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	930.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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